Determine if $\sum_{n=1}^{\infty} \frac{1}{\exp(1/n)}$is convergent or divergent

I have problem figuring out if this series is convergent or divergent. Please help!

$$\sum_{n=1}^{\infty} \frac{1}{\exp(1/n)}$$

Thanks

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2 Answers

I'd use the Divergence Test to show that this infinite series diverges.

The theorem states that if a series ${a_n}$ does not converge to $0$, then $\sum_{n=1}^{\infty} a_{n}$ diverges.

$$\lim_{n\to\infty}\frac{1}{e^{\frac{1}{n}}} = \frac{1}{e^0} \to 1 \ne0 \implies a_{n} \ \ \text {diverges}$$

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thanks! it helps – Jaden Q Nov 7 '12 at 6:01

as n goes to infinity , $e^{\frac{1}{n}}$ goes to 1. So your limit of summand goes to 1.. Therefore it should be divergent.

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Reread the OP. It's $e^{-\frac{1}{n}}$. – Zach L. Nov 7 '12 at 4:53
It's just the same result, @ZachL. – DonAntonio Nov 7 '12 at 4:56